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## Fundamenta Mathematicae

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## On topological groups admitting a base at the identity indexed by $\omega ^{\omega }$

### Tom 238 / 2017

Fundamenta Mathematicae 238 (2017), 79-100 MSC: Primary 22A05, 54H11; Secondary 06A06. DOI: 10.4064/fm188-9-2016 Opublikowany online: 24 February 2017

#### Streszczenie

A topological group $G$ is said to have a local $\omega ^\omega$-base if the neighbourhood system at the identity admits a monotone cofinal map from the directed set $\omega ^\omega$. In particular, every metrizable group is such, but the class of groups with a local $\omega ^\omega$-base is significantly wider. The aim of this article is to better understand the boundaries of this class, by presenting new examples and counter-examples. Ultraproducts and non-archimedean ordered fields lead to natural families of non-metrizable groups with a local $\omega ^\omega$-base which nevertheless are Baire topological spaces.

More examples come from such constructions as the free topological group $F(X)$ and the free Abelian topological group $A(X)$ of a Tychonoff (more generally uniform) space $X$, as well as the free product of topological groups. We show that 1) the free product of countably many separable topological groups with a local $\omega ^\omega$-base admits a local $\omega ^\omega$-base; 2) the group $A(X)$ of a Tychonoff space $X$ admits a local $\omega ^\omega$-base if and only if the finest uniformity of $X$ has an $\omega ^\omega$-base; 3) the group $F(X)$ of a Tychonoff space $X$ admits a local $\omega ^\omega$-base provided $X$ is separable and the finest uniformity of $X$ has an $\omega ^\omega$-base.

#### Autorzy

• Arkady G. LeidermanDepartment of Mathematics
Ben-Gurion University of the Negev
P.O.B. 653
Beer Sheva, Israel
e-mail
• Vladimir G. PestovDepartment of Mathematics and Statistics
University of Ottawa
585 King Edward Avenue
and
Departamento de Matemática
e-mail
• Artur H. TomitaInstituto de Matemática e Estatistica